Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case

Abstract

We obtain quantitative bounds in the polynomial Szemerédi theorem of Bergelson and Leibman, provided the polynomials are homogeneous and of the same degree. Such configurations include arithmetic progressions with common difference equal to a perfect kth power.

On the polynomial Szemerédi theorem in finite fields

“…By the same argument that appeared in [14], which uses the PET induction scheme introduced by Bergelson and Leibman in [2], we have the following proposition.…”

“…This can be proven by carrying out the argument in Sections 3-5 of [14] almost word-for-word, but in the finite field setting instead of the integer setting. In fact, the proof in finite fields is even simpler than this, since the variables in the definition of Λ P 1 ,...,Pm range over all of F q instead of over intervals of vastly different sizes as they do in [14].…”

Bounds for sets with no polynomial progressions

“…where 1 ≤ ≪ ℓ 1 for each ∈ ℓ−( +1) , by applying Lemma 5.7 once with ( 0 , 0 ) = ( , ) for each ∈ ℓ−( +1) and ≤ . Starting from (16) and repeating this implication ℓ − 2 more times gives the conclusion of the proposition.…”

Three-term polynomial progressions in subsets of finite fields

“…The case when m = 1 is covered by Sárközy's Theorem [15], which dealt with P 1 = y 2 , and later generalizations to other polynomials, such as work by Sárközy [16], Balog, Pelikán, Pintz, and Szemerédi [1], Slijepčević [18], and Lucier [13]. When m ≥ 2, the only quantitative result for progressions involving nonlinear polynomials is due to Prendiville [14], who dealt with the special case when P i = a i y d for a fixed d ∈ N.…”

“…This eventually leads to a bound for (5) in terms of an average of averages over linear configurations. For example, Prendiville [14] bounds (5) by an average of local Gowers U s -norms, where the degree s grows extremely quickly as the degrees of the P i and the length of the progression grow. The general strategy of the proof of Theorem 1.2 is also to use Cauchy-Schwarz to bound (6) E x,y∈Fq f 0 (x)f 1 (x + P 1 (y))f 2 (x + P 2 (y)) in terms of an average…”